The notions of the ``weak dimension'' \(\text{wdim}(P)\) and the ``chain-weak dimension'' \(\text{cwdim}(P)\) of a (finite) poset \(P\) are introduced. The chain-weak dimension is the smallest \(n\) such that \(P\) can be embedded in a product of \(n\) chain-weak orders, a chain-weak order being a poset \(P = C_1 \cup \cdots \cup C_m\), where each \(C_i\) is a disjoint union of chains and such that when \(i < j\) and \(x \in C_i\), \(y \in C_j\), one has \(x \leq y\) in \(P\). Weak orders are chain-weak orders in which the \(C_i\)'s are antichains; and \(\text{wdim}(P)\) is defined similarly in terms of these. It is shown that \(\text{wdim}(P)\) is essentially the same as the usual dimension of \(P\), the smallest \(n\) such that \(P\) embeds in a product of \(n\) chains. Properties of \(\text{cwdim}(P)\) are studied.